A mob that is both rare and elite has a silver dragon with a wing on its portrait.

A rare mob or monster is on a different timer than normal mobs in the area or sometimes only spawns X out of every Y instances you enter. Like elite mobs, rare mobs have a dragon around their portrait; however, unlike elites, this dragon icon is silver and lacks a wing on the side. They can also be elite (aka Rare Elite), in which case the wing is present. Rare mobs also have a unique name that distinguishes them from normal mobs, and unlike elite mobs (unless they are elite too), no "Rare" will appear after their name in the tooltip.

The loot dropped from rare mobs is unique and locates almost exclusively within the uncommon table and below it (Low level rare mobs will often drop common loot.) Rare mobs found in instances may also drop an item from the rare table, which will frequently bind on pickup. Rare mobs found in Outland will only drop rare loot, which are all bind on equip.

Rares are sometimes erroneously referred to as "silver elites." However, unlike elite mobs (which have a gold dragon around their portrait), rare mobs are generally not tougher than normal mobs of the same level. Rare mobs typically have comparable health and DPS of a mob their level, but tend to have an extra skill or two. This skill can often be guessed by their name. There are also some mobs that are both rare and elite. Previously, such mobs had a portrait identical to that of an elite mob (gold dragon with a wing), making it indistinguishable without consulting some outside source. As of patch 2.0.1, elite rare mobs have silver dragon, like regular rare mobs, but with a wing, like elite mobs.

Rare mobs are always named, although not all named mobs are rare. Rare mobs are never quest objectives, but some drop quest-starting items.

Rare mobs in the main game world have respawn times that are generally at longer intervals than normal mobs. Although it is difficult to determine the exact respawn rate of any given mob, some rares may take as many as several hours between spawns. As a result, some rares that are actively hunted by players may be very rarely seen, such as Time-Lost Proto Drake in Storm Peaks, which drops highly valued mount, [Reins of the Time-Lost Proto-Drake].

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Some rare mobs are also elite mobs, so they are both hard to find and at least 30% stronger than the typical normal mob of their level. Elite rares, as expected, should typically drop even better loot than a typical rare mob, but this is not always the case.

There are a number of tools available to assist in locating the rare mobs for the above achievements.
First, targetting macros can be created that will target the first mob found from a list of partial names entered. See Medium Rare and Northern Exposure for cut-and-paste versions of those macros.

There are also addons that assist with the task. (Note: these addons may not have been updated to the latest version of WoW; WoWwiki does not maintain these.)

NPCscan - automatically targets and alerts when one of the mobs it has been set to find comes into range.

SilverDragon - targets and alerts. Includes a database of known rares.

Once a target is acquired with one of the above addons, it can be difficult to identify in which direction the mob lies. A nice tool for addressing that is the Focus Lens (made by Jewelcrafting), which comes in three color variations: [Purple], [Green], and [both]. Using a focusing lens creates a beam between the user and his target (in this case, the rare mob).

One exception to this rule are Razorfen Spearhides who are guarding Overlord Ramtusk. Their portrait does occasionally appear with a silver dragon around it, but they are definitely not rare.

In actuality, there are several instances with mobs who are displayed with a silver dragon around it, yet spawn every time the instance is entered. This is because a patch changed them from a rare spawning mob into a normal spawning mob. Blackrock Depths has a number of examples of this, such as Pyromancer Loregrain.